Subgroups — C₅×#13528 #11936248

Order 1280.

378 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#367	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#366	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#357	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#357	C₂	1	1	yes	maximal, characteristic
5	C₅×#1970	C₂	1	1	yes	maximal, characteristic
6	C₅×#1846	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
9	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₅×#107	C₂²	1	1	yes	characteristic
12	C₂×C₅×#107	C₂²	1	1	yes	characteristic
13	C₂×C₅×#107	C₂²	1	1	yes	characteristic
14	C₂×C₅×#106	—	2	1	no	—
15	C₂×C₅×#102	—	2	1	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#390	—	2	1	no	—
18	C₅×#382	—	2	1	no	—
19	C₅×#367	—	2	2	no	—
20	C₅×#366	—	2	2	no	—
21	C₅×#357	—	2	2	no	—
22	C₅×#357	—	2	2	no	—
23	#13528	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×D₄	—	2	1	no	—
25	C₂×C₅×D₈	—	2	1	no	—
26	C₂×C₅×D₈	—	2	1	no	—
27	C₂×C₅×D₈	—	2	1	no	—
28	C₂×C₅×D₈	—	2	1	no	—
29	C₂×C₅×D₈	—	2	1	no	—
30	C₂×C₅×D₈	Q₈	1	2	yes	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₂×C₅×C₄⋊C₄	—	2	1	no	—
34	C₂×C₅×C₄⋊C₄	—	2	1	no	—
35	C₂×C₅×C₄⋊C₄	—	2	1	no	—
36	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₅×C₁₆	—	2	1	no	—
39	C₂×C₅×C₁₆	—	2	1	no	—
40	C₂×C₅×C₁₆	—	2	1	no	—
41	C₂×C₅×C₁₆	—	2	1	no	—
42	C₂×C₅×C₁₆	D₄	1	2	yes	—
43	C₂×C₅×C₁₆	D₄	1	2	yes	—
44	C₅×#107	—	2	1	no	—
45	C₅×#107	—	2	1	no	—
46	C₅×#107	—	2	1	no	—
47	C₅×#107	—	2	1	no	—
48	C₅×#107	—	2	1	no	—
49	C₅×#107	—	2	1	no	—
50	C₅×#106	—	4	2	no	—
51	C₅×#102	—	4	2	no	—
52	C₄×C₅×C₈	—	2	1	no	—
53	C₄×C₅×C₈	—	2	1	no	—
54	C₅×#95	—	4	2	no	—
55	C₂×#367	C₂×C₅	1	1	yes	characteristic, complemented
56	C₂×#366	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×#357	C₂×C₅	1	1	yes	characteristic
58	C₂×#357	C₂×C₅	1	1	yes	characteristic
59	#1970	C₂×C₅	1	1	yes	characteristic
60	#1846	C₂×C₅	1	1	yes	characteristic, complemented
61	#1374	C₂×C₅	1	1	yes	characteristic, complemented
62	C₂⁴×C₅	—	4	1	no	—
63	C₂×C₅×D₄	—	2	2	no	—
64	C₂×C₅×D₄	—	4	1	no	—
65	C₂×C₅×D₄	—	4	1	no	—
66	C₂×C₅×D₄	—	4	1	no	—
67	C₂×C₅×D₄	—	4	1	no	—
68	C₂×C₅×D₄	—	4	1	no	—
69	C₂²×C₄×C₅	—	4	1	no	—
70	C₂²×C₄×C₅	—	4	1	no	—
71	C₂²×C₄×C₅	—	4	1	no	—
72	C₂²×C₄×C₅	D₈	1	2	yes	—
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₅×D₈	—	2	2	no	—
75	C₅×D₈	—	2	2	no	—
76	C₅×D₈	—	4	1	no	—
77	C₅×D₈	—	4	1	no	—
78	C₂×C₅×C₈	—	2	1	no	—
79	C₂×C₅×C₈	—	2	1	no	—
80	C₂×C₅×C₈	D₈	1	2	yes	—
81	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₅×C₈	#41	1	1	yes	characteristic
85	C₂×C₅×C₈	#41	1	1	yes	characteristic
86	C₂×C₅×C₈	#41	1	1	yes	characteristic
87	C₅×C₄⋊C₄	—	4	1	no	—
88	C₅×C₄⋊C₄	—	4	1	no	—
89	C₅×C₄⋊C₄	—	4	1	no	—
90	C₅×C₄⋊C₄	—	4	1	no	—
91	C₅×C₄⋊C₄	—	4	1	no	—
92	C₅×C₄⋊C₄	—	4	1	no	—
93	C₅×C₂²⋊C₄	—	8	4	no	—
94	C₄²×C₅	—	2	1	no	—
95	C₄²×C₅	—	2	1	no	—
96	C₅×C₁₆	—	2	2	no	—
97	C₅×C₁₆	—	2	2	no	—
98	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
99	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
100	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
101	C₂×#107	C₂²×C₅	1	1	yes	characteristic
102	C₂×#107	C₂²×C₅	1	1	yes	characteristic
103	C₂×#107	C₂²×C₅	1	1	yes	characteristic
104	C₂×#106	—	2	1	no	—
105	C₂×#102	—	2	1	no	—
106	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
107	#390	—	2	1	no	—
108	#382	—	2	1	no	—
109	#367	—	2	2	no	—
110	#366	—	2	2	no	—
111	#357	—	2	2	no	—
112	#357	—	2	2	no	—
113	C₂³×C₅	—	4	2	no	—
114	C₂³×C₅	—	8	1	no	—
115	C₂³×C₅	—	8	1	no	—
116	C₂³×C₅	—	8	2	no	—
117	C₂³×C₅	—	8	2	no	—
118	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
119	C₅×D₄	—	4	2	no	—
120	C₅×D₄	—	4	2	no	—
121	C₅×D₄	—	8	1	no	—
122	C₅×D₄	—	8	1	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	8	1	no	—
127	C₂×C₄×C₅	—	8	1	no	—
128	C₂×C₄×C₅	—	8	1	no	—
129	C₂×C₄×C₅	—	8	2	no	—
130	C₂×C₄×C₅	—	8	2	no	—
131	C₂×C₄×C₅	—	8	2	no	—
132	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₅	#123	1	1	yes	characteristic
134	C₂×C₄×C₅	#127	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₅	#137	1	1	yes	characteristic
137	C₂×C₄×C₅	#137	1	1	yes	characteristic
138	C₅×C₈	—	2	2	no	—
139	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
140	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
141	C₅×C₈	#123	1	1	yes	characteristic
142	C₅×C₈	#123	1	1	yes	characteristic
143	C₂²×D₄	—	2	1	no	—
144	C₂×D₈	—	2	1	no	—
145	C₂×D₈	—	2	1	no	—
146	C₂×D₈	—	2	1	no	—
147	C₂×D₈	—	2	1	no	—
148	C₂×D₈	—	2	1	no	—
149	C₂×D₈	C₅×Q₈	1	2	yes	—
150	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
151	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₂²⋊C₄	—	4	2	no	—
156	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
157	C₂×C₁₆	—	2	1	no	—
158	C₂×C₁₆	—	2	1	no	—
159	C₂×C₁₆	—	2	1	no	—
160	C₂×C₁₆	—	2	1	no	—
161	C₂×C₁₆	C₅×D₄	1	2	yes	—
162	C₂×C₁₆	C₅×D₄	1	2	yes	—
163	#107	—	2	1	no	—
164	#107	—	2	1	no	—
165	#107	—	2	1	no	—
166	#107	—	2	1	no	—
167	#107	—	2	1	no	—
168	#107	—	2	1	no	—
169	#106	—	4	2	no	—
170	#102	—	4	2	no	—
171	C₄×C₈	—	2	1	no	—
172	C₄×C₈	—	2	1	no	—
173	#95	—	4	2	no	—
174	C₂²×C₅	—	8	2	no	—
175	C₂²×C₅	—	8	4	no	—
176	C₂²×C₅	—	16	2	no	—
177	C₂²×C₅	—	16	1	no	—
178	C₂²×C₅	—	16	1	no	—
179	C₂²×C₅	#474	1	1	yes	characteristic
180	C₂²×C₅	#480	1	1	yes	characteristic
181	C₂²×C₅	#493	1	1	yes	characteristic
182	C₂²×C₅	#508	1	2	yes	—
183	C₂²×C₅	#509	1	2	yes	—
184	C₄×C₅	—	4	2	no	—
185	C₄×C₅	—	16	1	no	—
186	C₄×C₅	—	16	1	no	—
187	C₄×C₅	—	16	1	no	—
188	C₄×C₅	#399	1	1	yes	characteristic
189	C₄×C₅	#474	1	1	yes	characteristic
190	C₄×C₅	#480	1	1	yes	characteristic
191	C₄×C₅	#497	1	1	yes	characteristic
192	C₂⁴	—	4	1	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	C₅×D₈	1	2	yes	—
203	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
204	D₈	—	2	2	no	—
205	D₈	—	2	2	no	—
206	D₈	—	4	1	no	—
207	D₈	—	4	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	C₅×D₈	1	2	yes	—
211	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
212	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
214	C₂×C₈	C₅×#41	1	1	yes	characteristic
215	C₂×C₈	C₅×#41	1	1	yes	characteristic
216	C₂×C₈	C₅×#41	1	1	yes	characteristic
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₄⋊C₄	—	4	1	no	—
222	C₄⋊C₄	—	4	1	no	—
223	C₂²⋊C₄	—	8	4	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₁₆	—	2	2	no	—
227	C₁₆	—	2	2	no	—
228	C₂×C₅	—	16	2	no	—
229	C₂×C₅	#2093	1	1	yes	characteristic
230	C₂×C₅	#2228	1	2	yes	—
231	C₂×C₅	#2234	1	2	yes	—
232	C₂×C₅	#2249	1	1	yes	characteristic
233	C₂×C₅	#2250	1	1	yes	characteristic
234	C₂³	—	4	2	no	—
235	C₂³	—	8	1	no	—
236	C₂³	—	8	1	no	—
237	C₂³	—	8	2	no	—
238	C₂³	—	8	2	no	—
239	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
240	D₄	—	4	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	8	1	no	—
243	D₄	—	8	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	8	1	no	—
248	C₂×C₄	—	8	1	no	—
249	C₂×C₄	—	8	1	no	—
250	C₂×C₄	—	8	2	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
254	C₂×C₄	C₅×#123	1	1	yes	characteristic
255	C₂×C₄	C₅×#127	1	1	yes	characteristic
256	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
257	C₂×C₄	C₅×#137	1	1	yes	characteristic
258	C₂×C₄	C₅×#137	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
261	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
262	C₈	C₅×#123	1	1	yes	characteristic
263	C₈	C₅×#123	1	1	yes	characteristic
264	C₅	#13528	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²	—	8	2	no	—
266	C₂²	—	8	4	no	—
267	C₂²	—	16	2	no	—
268	C₂²	—	16	1	no	—
269	C₂²	—	16	1	no	—
270	C₂²	C₅×#474	1	1	yes	characteristic
271	C₂²	C₅×#480	1	1	yes	characteristic
272	C₂²	C₅×#493	1	1	yes	characteristic
273	C₂²	C₅×#508	1	2	yes	—
274	C₂²	C₅×#509	1	2	yes	—
275	C₄	—	4	2	no	—
276	C₄	—	16	1	no	—
277	C₄	—	16	1	no	—
278	C₄	—	16	1	no	—
279	C₄	C₅×#399	1	1	yes	characteristic
280	C₄	C₅×#474	1	1	yes	characteristic
281	C₄	C₅×#480	1	1	yes	characteristic
282	C₄	C₅×#497	1	1	yes	characteristic
283	C₂	—	16	2	no	—
284	C₂	C₅×#2093	1	1	yes	characteristic
285	C₂	C₅×#2228	1	2	yes	—
286	C₂	C₅×#2234	1	2	yes	—
287	C₂	C₅×#2249	1	1	yes	characteristic
288	C₂	C₅×#2250	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₅×#367	C₂	semidirect
2. C₂×C₅×#366	C₂	semidirect
6. C₅×#1846	C₂	semidirect
7. C₅×#1374	C₂	semidirect
23. #13528	C₅	direct (N × H)
55. C₂×#367	C₂×C₅	semidirect
56. C₂×#366	C₂×C₅	semidirect
60. #1846	C₂×C₅	semidirect
61. #1374	C₂×C₅	semidirect
264. C₅	#13528	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#367	2
C₂×C₅×#366	2
C₂×C₅×#357	2
C₅×#1970	2
C₅×#1846	2
C₅×#1374	2
#13528	5

Subgroups — C5×#13528 #11936248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13528 #11936248